The two sides could not be separated at the end of the 90 minutes, with James Ward-Prowse's penalty cancelling out an opener from Josh Windass to leave the scores level at 1-1.
The Premier League side, who sacked Ralph Hasenhuttl earlier this week, managed to win the penalty shootout 6-5, though, with Ward-Prowse, Mohamed Elyounoussi, Ibrahima Diallo, Ainsley Maitland-Niles, Adam Armstrong and Theo Walcott all converting for the home team, and Dominic Iorfa's miss ultimately saw the visitors eliminated.
Wednesday were on the front foot in the early exchanges, with Fisayo Dele-Bashiru, Windass and Barry Bannan all having attempts inside the opening 18 minutes at St Mary's.
Bannan again tried his luck in the 22nd minute with an effort from distance, but the visitors managed to make the breakthrough in the 24th minute through Windass, who found the back of the net from a difficult angle.
Both sides had chances in the period that followed, with Joe Aribo and Sekou Mara forcing saves from Wednesday goalkeeper David Stockdale, while Callum Paterson and Tyreeq Bakinson had attempts for the League One outfit, with the latter forcing a smart save from Alex McCarthy.
Southampton managed to level the scores before the interval, though, with Ward-Prowse scoring from the penalty spot after Moussa Djenepo had been fouled by Marvin Johnson.
Adam Armstrong fired wide of the Wednesday goal early in the second period, with Southampton looking the more likely to score a second.
Stuart Armstrong came closest in the 87th minute when he hit the crossbar with a right-footed strike, before Duje Caleta-Car headed wide of the post in the final exchanges.
Ward-Prowse also had an opportunity in the fourth minute of added time at the end of the 90, but Stockdale made another smart save, with the match heading for penalties.
Both sides scored their opening five spot picks, but Wednesday missed their sixth through Iorfa, with Walcott's effort ultimately sending the Saints into the next round of the competition.